A short proof of the Goldberg-Seymour conjecture

For a multigraph G, ' (G) denotes the chromatic index of G, (G) the maximum degree of G, and (G) = \ 2|E (H) |{|V (H) |-1: H G and |V (H) | odd\}. As a generalization of Vizing's classical coloring result for simple graphs, the Goldberg-Seymour conjecture, posed in the 1970s, states that ' (G) =\ (G), (G) \ or ' (G) =\ (G) + 1, (G) \. Hochbaum, Nishizeki, and Shmoys further conjectured in 1986 that such a coloring can be found in polynomial time. A long proof of the Goldberg-Seymour conjecture was announced in 2019 by Chen, Jing, and Zang, and one case in that proof was eliminated recently by Jing (but the proof is still long) ; and neither proof has been verified. In this paper, we give a proof of the Goldberg-Seymour conjecture that is significantly shorter and confirm the Hochbaum-Nishizeki-Shmoys conjecture by providing an O (|V|⁵|E|³) time algorithm for finding a \ (G) + 1, (G) \-edge-coloring of G.